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Useful Map Properties: Distortion Pattern

Assessing and Measuring Distortion

Tissot's indicatrices in equatorial
Mollweide, Hammer and Eckert II maps. Owing to
limitations in my software, the indicatrices may be rendered
as ovals, or even irregular shapes; theoretical indicatrices
are always elliptical.

Every flat map includes some distortion of shape, area or length;
while some regions might be free of distortion, others could
suffer from severe error. Objectively assessing which regions are
affected and by how much is fundamental when choosing an
appropriate projection and aspect for a map.

Tissot's Indicatrix

A serious study of map projections usually involves a comparison
of how they are affected by the three main kinds of distortion —
area, shape and distance. Distortion may be visually estimated by
inspecting graticule patterns and the general shape of coastlines;
it can also be evaluated by measuring distances between selected
sets of points.
However, a systematic approach to quantitatively calculating
distortion had to wait for Nicolas A. Tissot's extremely
influential papers of 1878 and 1881 (some ideas were already
introduced in a work of 1859) presenting his ellipse of
distortion, universally known today as Tissot's
indicatrix.

Tissot imagined an infinitesimally small circle centered on some
point on the surface of the Earth, and considered its shape
after transformed by a given map projection. He proved it to be a
perfect ellipse, centered exactly on the corresponding point on
the map. Also,

if the projection is conformal at that point, the ellipse would
remain circular, albeit almost certainly larger or smaller than
the original, and maybe rotated

if instead the projection is equal-area at that point, the ellipse would
probably not be a circle, but have the same area as the original

if the projection is neither equal-area nor conformal at
that point, both shape and area would vary

Equatorial Mercator map, clipped at
85°N and 85°S, with Tissot's indicatrices (again,
the oversized circles here used for illustration are actually
ovals which slightly violate conformality)

The characteristic distortion pattern of a
projection may be roughly visualized by means of an array of
Tissot's indicatrices regularly spaced along the map. For some
projections, the angle and major and minor radii of each ellipse
can be calculated analytically. In practice, commonly the original
circles are simply numerically projected and rendered on the map;
in order to be visible — even after scaling — they must usually be
much larger than infinitesimal and don't necessarily look like
perfect ellipses.

All equal-area projections distort shape nearly everywhere. For
instance,
Hammer's
elliptical projection is not conformal, except, in the normal
aspect,
at the intersection of the Equator and the central meridian.
Mollweide's projection is only free
of shape distortion at the intersection of the central meridian
with the two standard parallels, at about 40° N and S.
Eckert's second projection
also has two standard parallels; if numerically projected, the
sharp break of angular deformation at the Equator creates some
peculiarly shaped indicatrices.

On the other hand, in
Mercator's conformal projection
all indicatrices remain circular in shape, parallels keep
parallelism, meridians are straight lines and always
perpendicular to every parallel. Areas are not preserved, but
greatly increase towards the top and bottom of the map:
circles at the poles would be infinitely large (this is to be
expected, since meridians cross one another on a sphere but
never touch in a Mercator map. Only infinite circles on
different meridians could be all concentric as in the globe's
poles).

Oblique Mercator map. Compared with the equatorial
version, the circles moved around but their relative sizes still
depend only on their distance from a horizontal line.

Another conformal projection, the azimuthal
stereographic preserves the shape of every circle,
naturally including indicatrices.

Scaling and Angular Deformations

Given a circle on Earth, Tissot considered pairs of diameters;
he proved there is always one pair which intersects orthogonally
(i.e., at a right angle) both on the circle's center and on its
projection in the map, where they comprise the ellipse's major
and minor axes. In addition, there is another pair intersecting
at a right angle on the circle, but maximally far from a right
angle on the map. This deviation is the maximum angular
deformation at that point and is, of course, zero if
the projection is conformal there.

Tissot developed equations relating the scale distortion (either
compression or stretching) at any point of a projection with the
maximum angular deformation: for conformality, for every pair of
orthogonal directions their scales must be the same;
for areal equivalence, they must be reciprocal. With his
formulas the deformation pattern may be calculated with any
desired precision and, when plotted on a map, directly evinces
zones of major areal or shape distortion.

On the left, equatorial
and oblique azimuthal orthographic maps; on the right, oblique
azimuthal stereographic map. All clipped to one hemisphere.

As shown by oblique orthographic and
Mercator maps, Tissot's indicatrices display an
overall deformation pattern, not affected by aspect
changes: in an azimuthal map, distortion depends only on the radial
distance from the conceptual center of the map; in a cylindrical
map, only on the vertical distance from the conceptual
"Equator". These are exactly the same patterns presented by the
normal versions.

Some Practical Examples

at least the horizontal scale is always exaggerated in
polar caps in every cylindrical map

having parallels and meridians crossing at right angles
everywhere is not enough for achieving conformality

Finally, indicatrices may help distinguishing between easily
confused projections, for instance three designs bounded by a
circle, ordinarily used in the equatorial aspect, with meridians
curved away from the Equator and extreme areal exaggeration near
the poles:

the van der
Grinten I is neither conformal nor equal-area;
indicatrices near the poles are evidence of shearing

in the van der
Grinten II projection meridians and parallels
intersect at right angles, thus there is no shearing; on the
other hand, away from the Equator the scale along meridians is
clearly larger than along parallels, therefore the map is not
conformal; obviously neither it is equivalent

in the simpler case of the
"Lagrange"
projection parallels and meridians are always orthogonal with
identical scale, and circles retain their shape: the
projection is conformal —
except at the poles, where the (ideal) indicatrices
become only halves of circles. This projection, as is typical
of conformal works, involves a broader range of scales: the
central portion of the map is noticeably smaller than in the
other two circular projections

Three projections bounded by
a circle: from left to right, van der Grinten I
(neither conformal nor equal-area, scaled down 61.5% in
comparison with other maps on this page), van der
Grinten II (ditto), "Lagrange" (conformal except at
poles, scaled down 48.2%)